|
| |
|
|
A154335
|
|
A triangular sequence of coefficients of polynomials: p(x,n)=(2*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).
|
|
0
| |
|
|
1, 1, 1, 1, 8, 1, 1, 35, 35, 1, 1, 126, 394, 126, 1, 1, 417, 3062, 3062, 417, 1, 1, 1324, 19895, 44680, 19895, 1324, 1, 1, 4111, 117021, 503827, 503827, 117021, 4111, 1, 1, 12602, 648616, 4882342, 9193838, 4882342, 648616, 12602, 1, 1, 38333, 3464840
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,5
|
|
|
COMMENTS
| Row sums are:
{1, 2, 10, 72, 648, 6960, 87120, 1249920, 20280960, 367960320,...}
|
|
|
FORMULA
| p(x,n)=(2*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}]
-(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).
Functional form:
p(x,n)=(2*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi[x, 1 - n, 1/2]
- (-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog[ -n, x]/x).
t(n,m)=Coefficients(p(x,n))
|
|
|
EXAMPLE
| {1},
{1, 1},
{1, 8, 1},
{1, 35, 35, 1},
{1, 126, 394, 126, 1},
{1, 417, 3062, 3062, 417, 1},
{1, 1324, 19895, 44680, 19895, 1324, 1},
{1, 4111, 117021, 503827, 503827, 117021, 4111, 1},
{1, 12602, 648616, 4882342, 9193838, 4882342, 648616, 12602, 1},
{1, 38333, 3464840, 42960752, 137516234, 137516234, 42960752, 3464840, 38333, 1}
|
|
|
MATHEMATICA
| Clear[p, x, n]; p[x_, n_] = (2*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}]
- (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]
|
|
|
CROSSREFS
| Sequence in context: A141686 A185412 A157148 * A142467 A142175 A142597
Adjacent sequences: A154332 A154333 A154334 * A154336 A154337 A154338
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 07 2009
|
| |
|
|
